, Volume 78, Issue 1, pp 342–377 | Cite as

Linear Rank-Width of Distance-Hereditary Graphs I. A Polynomial-Time Algorithm

  • Isolde Adler
  • Mamadou Moustapha Kanté
  • O-joung Kwon


Linear rank-width is a linearized variation of rank-width, and it is deeply related to matroid path-width. In this paper, we show that the linear rank-width of every n-vertex distance-hereditary graph, equivalently a graph of rank-width at most 1, can be computed in time \({\mathcal {O}}(n^2\cdot \log _2 n)\), and a linear layout witnessing the linear rank-width can be computed with the same time complexity. As a corollary, we show that the path-width of every n-element matroid of branch-width at most 2 can be computed in time \({\mathcal {O}}(n^2\cdot \log _2 n)\), provided that the matroid is given by its binary representation. To establish this result, we present a characterization of the linear rank-width of distance-hereditary graphs in terms of their canonical split decompositions. This characterization is similar to the known characterization of the path-width of forests given by Ellis, Sudborough, and Turner [The vertex separation and search number of a graph. Inf. Comput., 113(1):50–79, 1994]. However, different from forests, it is non-trivial to relate substructures of the canonical split decomposition of a graph with some substructures of the given graph. We introduce a notion of ‘limbs’ of canonical split decompositions, which correspond to certain vertex-minors of the original graph, for the right characterization.


Rank-width Linear rank-width Distance-hereditary graphs Vertex-minors Matroid branch-width Matroid path-width 



The authors would like to thank Sang-il Oum for pointing out that the computation of the path-width of matroids of branch-width at most 2 can be obtained as a corollary of our main result.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Isolde Adler
    • 1
  • Mamadou Moustapha Kanté
    • 2
  • O-joung Kwon
    • 3
    • 4
  1. 1.School of ComputingUniversity of LeedsLeedsUK
  2. 2.Université Clermont Auvergne, Université Blaise Pascal, LIMOS, CNRSAubièreFrance
  3. 3.Department of Mathematical SciencesKAISTYuseong-gu, DaejeonSouth Korea
  4. 4.Institute for Computer Science and ControlHungarian Academy of Sciences (MTA SZTAKI)BudapestHungary

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